I have a real-valued single-variable cadlag function $f$. I also have an increasing sequence of bounded real numbers $(t_k)_k$ that converges to some finite limit, say $t$. By the existence of left limits of $f$, I have $\lim_{k\to\infty}f(t_k) =: f(t-) < \infty$.
What I would like to know is how this property contradicts the statement $$\lvert f(t_k) - f(t_k-)\rvert \geq \varepsilon \qquad k = 0,1,\ldots$$ where $f(t_k-) := \lim_{s\uparrow t_k} f(s)$ and $\varepsilon > 0$.
By the existence of the left limit, I can write $$\lvert f(t_k) - f(t_-)\rvert < \varepsilon $$ for all $k\geq N$ for some $N \in \mathbb{N}$.
This seems like a contradiction to the statement above but I don't have the actual argument to show that.